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Improving Your CASH Flow: the Computer Algebra Shell

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Improving Your CASH Flow: the Computer Algebra Shell Improving your CASH flow: The C omputer Algebra SH ell Christopher Brown1, Hans-Wolfgang Loidl2, Jost Berthold3, and Kevin Hammond1 1 School of Computer Science, University of St. Andrews, UK. fchrisb,[email protected] 2 School of Mathematical and Computer Sciences, Heriot-Watt University, UK. [email protected] 3 DIKU Department of Computer Science, University of Copenhagen, Denmark. [email protected] Abstract. This paper describes CASH (the Computer Algebra SHell), a new interface that allows Haskell programmers to access the complete functionality of a number of computer algebra systems directly and in- teractively. Using CASH, Haskell programmers can access previously- unavailable mathematical software, and, in the reverse direction, users of computer algebra systems can exploit the rapidly growing Haskell code base and its rich set of libraries. In particular, CASH provides a simple and effective interface for users of computer algebra systems to parallelise their algorithms using domain-specific skeletons written in Haskell. 1 Introduction Users of functional programming languages can often feel as if they are in a ghetto. While foreign-function interfaces often exist to e.g. C, they can, indeed, feel foreign to use, especially where complex data structures are involved. And while library developers work hard to provide new functionality to functional programmers, this hard-won capability often bit-rots faster than it can be used effectively. As part of an EU-funded project4, we have been working to address some of these issues (and, incidentally, to provide a new user base for functional languages). In this way, we hope to open the gates to the ghetto of functional programming, breaking down some of the barriers that prevent wider use. This paper describes a new way to link Haskell [1] with computer algebra systems. CASH (the Computer Algebra SHell) provides a two-way interface to a number of widely-used computer algebra systems, including GAP [2], KANT [3], Mu- PAD[4], Maple [5] and MacAulay [6]. In order to achieve this, it exploits the generic SCSCP [7] protocol that has been implemented for all these systems, which, in turn, uses the OpenMath [8] standard format to describe the structure of mathematical objects. 4 SCIEnce: Symbolic Computation Infrastructure in Europe, RII3-CT-2005-026133. Interfacing in this way between (an interactive shell for) Haskell and com- puter algebra systems has several advantages. Firstly, these systems are domain- specific environments that provide a large number of specialised data-types and that build on decades of expert experience. For instance, the open source sys- tem GAP [2] comes with a library of complex computer algebra algorithms that have been contributed by hundreds or thousands of domain experts. Exploiting existing, optimised, environments such as GAP from within Haskell increases programmer productivity | the Haskell programmer does not have to learn the mathematics behind the code. Moreover, the bidirectional SCSCP interface also allows advanced functional language features to be used directly from a computer algebra system. For example, Haskell's parallelisation capabilities can be exploited without the user of the computer algebra system needing to know anything about either parallelism or functional programming. 1.1 Contributions The main technical contributions made by this paper are: 1. we show how CASH can be used to integrate Haskell and computer algebra code using SCSCP and OpenMath, giving examples that show how CASH can be used to call computer algebra systems from Haskell and vice-versa; 2. we show how to define domain-specific parallel skeletons for computer algebra in the Eden [9] dialect of Haskell, in particular, we describe a new multiple homomorphic images skeleton; and 3. we expose these parallel skeletons as SymGrid-Par [10] services, that can be called directly both from the CASH shell and from within the shell of an SCSCP-enabled computer algebra client. The idea of interfacing functional languages and computer algebra systems is, of course, not new. For example, GPH-Maple [11] and Eden-Maple [12] pro- vide dedicated sequential and parallel interfaces, respectively, to the commercial Maple system. However, CASH goes beyond these earlier approaches in terms of both usability and generality: CASH uses a Haskell implementation of the standardised OpenMath-based SCSCP [7] interface, which means that it can in- teract with any computer algebra system that implements the SCSCP standard; moreover, it also allows any SCSCP-compliant system to interact with Haskell. 2 Linking Haskell and Computer Algebra Systems CASH uses the GHCi interpreter to provide a Haskell-side shell for accessing computer algebra systems via SCSCP. The underlying libraries provide data structures and conversion functions targeted at computer algebra, plus a \com- mand line API" for an interactive front end, that is used to establish a connec- tion to, and interact with, an SCSCP-enabled system. To show how powerful, expressive and useful the CASH system can be, we will consider a simple exam- ple involving matrix groups over finite fields. We first define two matrices, m1 and m2 over the finite field Z3 = f0; 1; 2g, interactively, using the CASH shell *Cash> let m1 = [[Z(3)^0, Z(3)^0],[Z(3),0*Z(3)]] *Cash> let m2 = [[Z(3), Z(3)],[Z(3),0*Z(3)]] Here we have used the GAP-like syntax Z(3) for the element which generates the multiplicative group in the finite field Z3, 0*Z(3) for the 0-element and Z(3)^0 for the 1-element. Using CASH, we can directly exploit any of the GAP functions on group algebra. Here we compute the multiplicative matrix groups generated by each of these two matrices and then determine their intersection. *Cash> group(m1) Group([ [ [ Z(3)^0, Z(3)^0 ], [ Z(3), 0*Z(3) ] ] ]) *Cash> group(m2) Group([ [ [ Z(3), Z(3) ], [ Z(3), 0*Z(3) ] ] ]) *Cash> intersection(group(m1), group(m2)) [ [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ], [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ] The result of the intersection is given by two matrices, the first of which is the unit matrix and the second of which is the unit matrix multiplied by the scalar Z(3). Note that the result of each group operation is simply a symbolic representation of the result in terms of one or more generators, rather than a enumeration of all possible elements of the group. This is important, because the number of elements in a group can be huge, and the user is usually only interested in some specific properties of the group. We can often determine these properties without needing to generate all the elements of the group. This dramatically reduces the amount of computation that is required. 2.1 The SCSCP Interface The interface between a client and a server is defined by the SCSCP communi- cation protocol [7], which itself builds on the existing OpenMath standard [8] for data representation of mathematical objects. The SCSCP communication pro- tocol defines how messages are exchanged between client and server processes. Its main functionality is to support (remote) calls of SCSCP server-side services and to deliver results to the SCSCP client. In order to support automatic ser- vice discovery, the protocol also specifies how to obtain a list of all the available services that are supported by the SCSCP server, including their types. Figure 1 shows a CASH session interacting with a GAP server using SCSCP. The CASH session first connects to GAP with a Procedure Call message, containing information such as the Procedure name, any Arguments and any Options/Attributes, encoded in OpenMath. Alternatively, the user may inter- rupt the interaction at any time between the computer algebra system and the server. In this case the computation terminates and no result is returned. The GAP server responds to the CASH client by sending a Procedure Completed message: an OpenMath encoding containing the information about the result of the procedure. This message contains the (OpenMath) Result value; any Manda- tory Additional Information, such as the internal SCSCP call identifier; and any Optional additional information, such as the procedure runtime/memory usage. CASH GAP Procedure Call "WS_Group", OMObj, "CASH", Return to sender [Interrupt Signal] [Procedure Compeleted] OMOBJ Result, "GAP" [Procedure Terminated] Error, "GAP" Fig. 1. An example interaction between CASH and GAP using SCSCP Alternatively, the GAP server may instead return a Procedure Terminated message containing an Error message. The CASH user-level interface is provided by the callSCSCP function, which takes a service name and a list of OpenMath objects as arguments and issues an SCSCP call to the server. To simplify this interface, a family of functions call1, call2 etc. are defined, which hide the (un)marshalling of the data-structures. 2.2 The OpenMath Data Format The OpenMath [8] data format is an XML-based representation of mathematical objects such as polynomials, finite field elements or permutations. Formally, an OpenMath object is a labelled tree whose leaves form typical computer algebra type representations. These include integers, unicode strings, variables or sym- bols, where symbols consist of a name and a content dictionary. For instance, a vector of finite field elements [[Z(3)^0,Z(3)^0]] can be encoded in the Open- Math representation shown in Figure 2. This uses the linalg2 content dictionary to define matrix and matrixrow. 3 Calling GAP from Haskell and vice-versa We now give two examples of using CASH that show the usefulness of being able to call computer algebra functionality from a Haskell environment without having to step out of the functional paradigm. <OMOBJ> <OMA> <OMS cd="linalg2" name="matrix"/> <OMA> <OMS cd="linalg2" name="matrixrow"/> <OMA> <OMS cd="arith1" name="power"/> <OMA> <OMS cd="finfield1" name="primitive_element"/> <OMI>3</OMI> </OMA> <OMI>0</OMI> </OMA> <OMA> <OMS cd="arith1" name="power"/> <OMA> <OMS cd="finfield1" name="primitive_element"/> <OMI>3</OMI> </OMA> <OMI>0</OMI> </OMA> </OMA> </OMA> </OMOBJ> Fig.
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